Hypergeometric functions

------------------------



Common hypergeometric series

............................



For convenience, most of the hypergeometric series of low order are

provided as standalone functions. They can equivalently be evaluated using

:func:`hyper`. As will be demonstrated further in the individual docstrings,

all the following functions implement analytic continuations and/or asymptotic

expansions with respect to the argument `z`, thereby permitting evaluation

anywhere in the complex plane. Functions of higher degree can be computed

via :func:`hyper`, but generally only in rapidly convergent instances.



:func:`hyp0f1`

^^^^^^^^^^^^^^

.. autofunction:: mpmath.functions.hyp0f1(a, z)



:func:`hyp1f1`

^^^^^^^^^^^^^^

.. autofunction:: mpmath.functions.hyp1f1(a, b, z)



:func:`hyp1f2`

^^^^^^^^^^^^^^

.. autofunction:: mpmath.functions.hyp1f2(a1, b1, b2, z)



:func:`hyp2f0`

^^^^^^^^^^^^^^

.. autofunction:: mpmath.functions.hyp2f0(a, b, z)



:func:`hyp2f1`

^^^^^^^^^^^^^^

.. autofunction:: mpmath.functions.hyp2f1(a, b, c, z)



:func:`hyp2f2`

^^^^^^^^^^^^^^

.. autofunction:: mpmath.functions.hyp2f2(a1, a2, b1, b2, z)



:func:`hyp2f3`

^^^^^^^^^^^^^^

.. autofunction:: mpmath.functions.hyp2f3(a1, a2, b1, b2, b3, z)



Generalized hypergeometric functions

....................................



:func:`hyper`

^^^^^^^^^^^^^^

.. autofunction:: mpmath.functions.hyper(a_s, b_s, z)



:func:`hypercomb`

^^^^^^^^^^^^^^^^^

.. autofunction:: mpmath.functions.hypercomb



Confluent U and Whittaker functions

...................................



:func:`hyperu`

^^^^^^^^^^^^^^

.. autofunction:: mpmath.functions.hyperu(a, b, z)



:func:`whitm`

^^^^^^^^^^^^^^

.. autofunction:: mpmath.functions.whitm(k,m,z)



:func:`whitw`

^^^^^^^^^^^^^^

.. autofunction:: mpmath.functions.whitw(k,m,z)



Meijer G-function

...................................



:func:`meijerg`

^^^^^^^^^^^^^^^^^

.. autofunction:: mpmath.functions.meijerg(a_s,b_s,z,r=1,**kwargs)



Appell hypergeometric function of two variables

...............................................



:func:`appellf1`

^^^^^^^^^^^^^^^^^

.. autofunction:: mpmath.functions.appellf1(a,b1,b2,c,x,y)